Explanation and Prediction: An Architecture for Default and Abductive Reasoning

40 Pages·2007·0.27 MB·English

Explanation and Prediction(cid:0) An
Architecture for Default and Abductive
Reasoning
David Poole
Department of Computer Science(cid:0)
The University of British Columbia(cid:0)
Vancouver(cid:0) B(cid:1)C(cid:1)(cid:0) Canada V(cid:2)T (cid:3)W(cid:4)
(cid:5)(cid:2)(cid:6)(cid:7)(cid:8) (cid:9)(cid:9)(cid:10) (cid:2)(cid:9)(cid:4)(cid:7)
poole(cid:11)cs(cid:1)ubc(cid:1)ca
September (cid:3)(cid:4)(cid:0) (cid:3)(cid:12)(cid:12)(cid:13)
Abstract
Although there are many arguments that logic is an appropriate
tool forarti(cid:0)cial intelligence(cid:1) there hasbeen a perceived problem with
the monotonicity of classical logic(cid:2) This paper elaborates on the idea
that reasoning should be viewed as theory formation where logic tells
us the consequences of our assumptions(cid:2) The two activities of pre(cid:3)
dicting what is expected to be true and explaining observations are
considered inasimple theoryformationframework(cid:2) Propertiesofeach
activity are discussed(cid:1) along with a number of proposals as to what
should be predicted or accepted as reasonable explanations(cid:2) An ar(cid:3)
chitecture is proposed tocombine explanation and prediction into one
coherentframework(cid:2) Algorithmsused toimplement thesystemaswell
as examples from a running implementation are given(cid:2)
Key words(cid:4) defaults(cid:1) conjectures(cid:1) explanation(cid:1) prediction(cid:1) abduc(cid:3)
tion(cid:1) dialectics(cid:1) logic(cid:1) nonmonotonicity(cid:1) theory formation
(cid:0)
Explanation and Prediction (cid:1)
(cid:0) Introduction
One way to do research in Arti(cid:2)cial Intelligence is to argue that we need a
certain number of tools and to augment these only when they have proven
inadequate for some task(cid:3) In this way we can argue we need at least the (cid:2)rst
order predicate calculus to reason about individuals and relations among
individuals (cid:4)given that we want to indirectly describe individuals(cid:5) as well
as describe the conjunction(cid:5) disjunction and negation of relations(cid:6) (cid:7)Hayes(cid:8)(cid:8)(cid:5)
Moore(cid:9)(cid:1)(cid:5) Genesereth(cid:9)(cid:8)(cid:10)(cid:3)
Non(cid:11)monotonicity has often been cited as a problem with using logic as
a basis for commonsense reasoning(cid:3) In (cid:7)PGA(cid:9)(cid:8)(cid:10) it was argued that instead
of deduction from our knowledge(cid:5) reasoning should be viewed as a process of
theory formation(cid:3) In (cid:7)Poole(cid:9)(cid:9)a(cid:10) it was shown how default reasoning can be
viewed in this way by treating defaults as possible hypotheses that can be
used in an explanation(cid:3)
It has also been recognised (cid:4)e(cid:3)g(cid:3)(cid:5)(cid:7)Charniak(cid:9)(cid:12)(cid:5) PGA(cid:9)(cid:8)(cid:5) Cox(cid:9)(cid:8)(cid:5)Reggia(cid:9)(cid:13)(cid:10)(cid:6)
that abduction is an appropriate way to view diagnostic and recognition
tasks(cid:3) In diagnosis(cid:5) for example(cid:5) the diseases and malfunctions are the pos(cid:11)
sible hypotheses that can be used to explain some observations(cid:3)
We can argue we want to use logic and do hypothetical reasoning(cid:3) This
research considers the simplest form of hypothetical reasoning(cid:5) namely the
case where the user provides a set of possible hypotheses they are prepared
to accept as part of a theory(cid:3) This is the framework of the Theorist system
(cid:7)Poole(cid:9)(cid:9)a(cid:5) PGA(cid:9)(cid:8)(cid:10)(cid:3) The distinctions outlined in this paper were found from
experience by using the system(cid:5) explaining to others how to use the system
and in building applications (cid:7)Poole(cid:9)(cid:8)b(cid:10)(cid:3)
(cid:0)(cid:1)(cid:0) Theorist Framework
We assume we are given a standard (cid:2)rst order language over a countable
alphabet (cid:7)Enderton(cid:8)(cid:1)(cid:10)(cid:3) By a formula we mean a well formed formula in
this language(cid:3) By an instance of a formula we mean a substitution of terms
in this language for free variables in the formula(cid:3) In this paper the Prolog
convention of variables starting with an upper case letter is used(cid:3)
The framework (cid:7)Poole(cid:9)(cid:9)a(cid:10) is de(cid:2)ned in terms of two sets of formulae(cid:14)
A is a set of closed formulae which we are taking as given(cid:5) and
Explanation and Prediction (cid:13)
H is a set of (cid:4)possibly open(cid:6) formulae which we take as the (cid:15)possible hy(cid:11)
potheses(cid:16)(cid:3)
De(cid:0)nition (cid:1)(cid:2)(cid:1) A scenario of (cid:4)A(cid:0)H(cid:6) is a set D of ground instances of
elements of H such that D (cid:0)A is consistent(cid:0)
A scenario is a set of hypotheses that could be true based on what we are
given(cid:3)
De(cid:0)nition (cid:1)(cid:2)(cid:3) If g is a closed formula(cid:1) an explanation of g from (cid:4)A(cid:0)H(cid:6)
is a scenario of (cid:4)A(cid:0)H(cid:6) which (cid:2)together with A(cid:3) implies g(cid:0)
ThusD isan explanationof g from(cid:4)A(cid:0)H(cid:6)ifD isa setofground instances
of elements of H such that D (cid:0)A is consistent and D (cid:0)A j(cid:17) g(cid:3)
De(cid:0)nition (cid:1)(cid:2)(cid:4) An extension of (cid:4)A(cid:0)H(cid:6) is the set of logical consequences of
A together with a maximal (cid:2)with respect to set inclusion(cid:3) scenario of (cid:4)A(cid:0)H(cid:6)(cid:0)
The following theorem was proved in (cid:7)Poole(cid:9)(cid:9)a(cid:10) and follows from the
compactness theorem of the (cid:2)rst order predicate calculus (cid:7)Enderton(cid:8)(cid:1)(cid:10)(cid:3)
Theorem (cid:1)(cid:2)(cid:5) There is an explanation of g from A(cid:0)H i(cid:4) g is in some ex(cid:5)
tension of A(cid:0)H(cid:0)
In (cid:7)Poole(cid:9)(cid:9)a(cid:10) it was shown that (cid:1) (cid:1) H corresponds exactly to the normal
default (cid:14) (cid:1)(cid:2)(cid:1) of (cid:7)Reiter(cid:9)(cid:18)(cid:10)(cid:3) It was also argued that the extra power of Reiter(cid:19)s
defaults was not needed(cid:3) Both (cid:7)Reiter(cid:9)(cid:18)(cid:10) and (cid:7)Poole(cid:9)(cid:9)a(cid:10) showed how their
systems can be used for default reasoning(cid:5) but not what such reasoning was
for(cid:3)
(cid:0)(cid:1)(cid:2) Explanation and Prediction
There are two activities we will consider(cid:5) namely explaining observations
and predicting what is expected to be true(cid:3) These are both considered to be
instances of the Theorist framework(cid:3)
Imaketheassumptionthat wedo notneedmorethantheTheoristframe(cid:11)
work(cid:3) This may turn out to be incorrect(cid:5) but if it is(cid:5) we will have found a
good reason to add extra features to our system(cid:3) I make no a priori as(cid:11)
sumption that the same hypotheses should be used both for explanation and
Explanation and Prediction (cid:20)
prediction(cid:21) in fact there are good reasons for not making them the same(cid:3) If
we later (cid:2)nd out they coincide(cid:5) again we will have learnt something(cid:3)
(cid:0)
As such(cid:5) the following sets of formulae are provided by the user(cid:14)
F is the set of facts(cid:5) which are taken as being true of the domain(cid:21)
(cid:22) is the set of defaults(cid:5) possible hypotheses which can be used in prediction(cid:21)
(cid:23) is the setof conjectures(cid:5) possibly hypotheses whichcan be used in explain(cid:11)
ing observations(cid:21)
O is the set of observations that have been made about the actual world(cid:3)
(cid:1) Prediction
A problem many people in AI have been working on is the problem of pre(cid:11)
dicting what one expects to be true in some (cid:4)real or imaginary(cid:6) world based
on the information one has about that world(cid:3)
The most conservative form of prediction is logical consequence from our
knowledge(cid:3) If axioms A are true in some world(cid:5) any logical consequence of
A must also be true in that world(cid:3) This is the essence of classical logic(cid:3)
Many people have argued that such a notion is too weak for common
sense prediction(cid:21) sometimes we want to make assumptions as to what we
expectto be true(cid:3) This is the basis of muchwork on nonmonotonic reasoning
(cid:7)Bobrow(cid:9)(cid:18)(cid:10)(cid:3)
We consider defaults as assumptions one is prepared to make about the
world(cid:5) unless they can be shown to be wrong(cid:3) In the Theorist framework(cid:5)
defaults are possible hypotheses used for prediction (cid:7)Poole(cid:9)(cid:9)a(cid:10)(cid:3)
What should be predicted based on such hypothetical reasoning seems to
be uncontroversial if there are no con(cid:24)icting defaults (cid:4)i(cid:3)e(cid:3)(cid:5) there is only one
extension(cid:6)(cid:3) In this section(cid:5) we discuss what should be predicted when there
are con(cid:24)icting defaults(cid:3)
We assume there is no other information on which to base our decision
(cid:4)e(cid:3)g(cid:3)(cid:5) speci(cid:2)city (cid:7)Poole(cid:9)(cid:12)(cid:10)(cid:5) probability (cid:7)Neufeld(cid:9)(cid:8)(cid:10)(cid:5) temporal considerations
(cid:7)Goebel(cid:9)(cid:8)(cid:10)(cid:6)(cid:3)
(cid:0)
As far as the preceding semantics are given(cid:0) the possible hypotheses(cid:0) H(cid:0) willin some
cases be (cid:1) and in some cases (cid:2)(cid:0)(cid:1)(cid:3) the given A will sometimes be F and sometimesF
together with an explanation of the observations(cid:4)
Explanation and Prediction (cid:12)
(cid:1)
Example (cid:3)(cid:2)(cid:1) Consider the following example
H (cid:17) f republican(cid:4)X(cid:6) (cid:2) hawk(cid:4)X(cid:6)(cid:0)
quaker(cid:4)X(cid:6) (cid:2) dove(cid:4)X(cid:6)(cid:0)
hawk(cid:4)X(cid:6) (cid:2) support(cid:5)star(cid:5)wars(cid:4)X(cid:6)(cid:0)
hawk(cid:4)X(cid:6) (cid:2) politically(cid:5)motivated(cid:4)X(cid:6)(cid:0)
dove(cid:4)X(cid:6) (cid:2) politically(cid:5)motivated(cid:4)X(cid:6)
quaker(cid:4)X(cid:6) (cid:2) religious(cid:4)X(cid:6)g
F (cid:17) f (cid:3)X (cid:4)(cid:4)dove(cid:4)X(cid:6)(cid:5)hawk(cid:4)X(cid:6)(cid:6)(cid:0)
quaker(cid:4)dick(cid:6)(cid:0)
republican(cid:4)dick(cid:6) g
Based on the above facts and defaults(cid:5) there are questions as to which of
the following should be predicted(cid:14)
dove(cid:4)dick(cid:6)
hawk(cid:4)dick(cid:6)
dove(cid:4)dick(cid:6)(cid:6)hawk(cid:4)dick(cid:6)
dove(cid:4)dick(cid:6)(cid:5)hawk(cid:4)dick(cid:6)
support(cid:5)star(cid:5)wars(cid:4)dick(cid:6)
politically(cid:5)motivated(cid:4)dick(cid:6)
religious(cid:4)dick(cid:6)
The rest of this section discusses four proposals of what should be pre(cid:11)
dicted(cid:3) There are based on the answers to the following question(cid:14)
If we have an explanation for p and and an explanation for q(cid:5) but we
know both cannot be true (cid:4)i(cid:3)e(cid:3)(cid:5) F j(cid:17) (cid:4)(cid:4)p(cid:5)q(cid:6)(cid:6)(cid:5) what should we predict(cid:25)
(cid:0)(cid:3) Either p or q but not both(cid:3)
(cid:1)(cid:3) Neither p nor q(cid:3)
(cid:13)(cid:3) p(cid:6)q(cid:3)
(cid:20)(cid:3) Nothing(cid:21) we have detected an inconsistency in our knowledge base(cid:3)
The following sections consider the consequences of each choice(cid:3)
(cid:1)
ThisexampleisbasedonanexamplebyMattGinsberg(cid:0)whichisbasedonanexample
of Ray Reiter(cid:4)
Explanation and Prediction (cid:26)
(cid:2)(cid:1)(cid:0) Predict if explainable
The (cid:2)rst de(cid:2)nition of prediction where we predict p or predict q corresponds
(cid:2)
to predictingwhateverisexplainable(cid:4)predictingwhat isin someextension (cid:6)(cid:3)
In example (cid:1)(cid:3)(cid:0)(cid:5) we would predict either
politically(cid:5)motivated(cid:4)dick(cid:6)(cid:5)hawk(cid:4)dick(cid:6)(cid:5)supports(cid:5)star(cid:5)wars(cid:4)dick(cid:6)
or
politically(cid:5)motivated(cid:4)dick(cid:6)(cid:5)dove(cid:4)dick(cid:6)
but not both(cid:3) This can be claimed to be reasonable because we were told
we could assume Dick is a hawk given no evidence to the contrary (cid:4)the only
evidencetothecontrarybeinganinternalinconsistency(cid:6)(cid:5)andsocanconclude
he is politically motivated and supports star wars(cid:3) We can also assume he is
a dove and so is politically motivated(cid:3) We just cannot assume he is both a
dove and hawk(cid:5) as this is inconsistent(cid:3)
This has the peculiar property that we both predict hawk(cid:4)dick(cid:6) and
predict (cid:4)hawk(cid:4)dick(cid:6) (cid:4)although in di(cid:27)erent extensions(cid:6)(cid:3) The following shows
this turns out to be general(cid:3)
Theorem (cid:3)(cid:2)(cid:3) There are multiple extensions if and only if there is some (cid:3)
such that (cid:3) is explainable and (cid:4)(cid:3) is explainable(cid:0)
Proof(cid:6) Suppose there are two extensions(cid:5) E(cid:0) and E(cid:1)(cid:3) Di(cid:27)erent
extensionsaremutuallyinconsistent(cid:5)soF(cid:0)E(cid:0)(cid:0)E(cid:1) isinconsistent(cid:3)
By the compactness of the (cid:2)rst order predicate calculus(cid:5) there
are (cid:2)nite subsets D(cid:0) and D(cid:1) of E(cid:0) and E(cid:1) respectively such that
F (cid:0)D(cid:0)(cid:0)D(cid:1) is inconsistent(cid:3) D(cid:0) is such an (cid:3) as D(cid:0) is in extension
E(cid:0) and (cid:4)D(cid:0) is in E(cid:1) (cid:4)as F (cid:0)D(cid:1) j(cid:17) (cid:4)D(cid:0)(cid:6)(cid:3)
Conversely(cid:5) suppose (cid:3) is explained by D(cid:0) and (cid:4)(cid:3) is explained
by D(cid:1)(cid:3) Extend D(cid:0) to extension E(cid:0) and D(cid:1) to E(cid:1)(cid:3) E(cid:0) and E(cid:1)
are mutually inconsistent(cid:5) and so are di(cid:27)erent(cid:3) Thus there are
(cid:0)
multiple extensions(cid:3)
As it seems wrong to both predict (cid:3) and predict (cid:4)(cid:3)(cid:5) membership in an
extension seems like a strange notion of prediction(cid:3) It corresponds more to
(cid:15)may be true(cid:16) than to prediction(cid:3)
(cid:2)
(cid:5)Reiter(cid:6)(cid:7)(cid:8) uses membership in one extension(cid:0) but does not claim that he is formal(cid:9)
ising prediction(cid:0) but rather (cid:10)an acceptable set of beliefs that one may hold about an
incompletelyspeci(cid:11)ed world(cid:12) (cid:5)Reiter(cid:6)(cid:7)(cid:0) p(cid:4) (cid:6)(cid:6)(cid:8)(cid:4)
Explanation and Prediction (cid:8)
(cid:2)(cid:1)(cid:2) Incontestable Scenarios
When both p and q can be explained(cid:5)but are mutuallyinconsistent(cid:5) it seems
reasonable to predict neither(cid:21) we were told we could assume p given no
evidence to the contrary(cid:5) but q is evidence to the contrary(cid:5) so we should
not assume p(cid:3)
This notion of prediction(cid:5) corresponds to predicting what can be ex(cid:11)
plained using an (cid:15)incontestable scenario(cid:16)(cid:3) This is a very sceptical form of
prediction where we predict some goal only if we have an argument why
the goal should be true (cid:4)i(cid:3)e(cid:3)(cid:5) the goal in explainable(cid:6) and we cannot (cid:2)nd an
argument why the argument for the goal should not be true(cid:3)
In example (cid:1)(cid:3)(cid:0)(cid:5) of the conclusions suggested only religious(cid:4)dick(cid:6) is pre(cid:11)
dicted(cid:3) We can(cid:19)t assume he is hawk since(cid:5) as far as we know(cid:5) he could be a
dove(cid:5) and we can(cid:19)t assume he is a dove(cid:5) as he may as well be hawk (cid:4)and he
can(cid:19)t be both(cid:6)(cid:5) so nothing that depends on these is predicted(cid:3)
De(cid:0)nition (cid:3)(cid:2)(cid:4) Scenario D of (cid:4)F(cid:0)(cid:22)(cid:6) is an incontestable scenario if (cid:4)D
is not explainable from (cid:4)F(cid:0)(cid:22)(cid:6)(cid:3)
The following lemma shows that being in an incontestable scenario is a
local property of instances of defaults and does not depend on other defaults
in an explanation(cid:3)
Lemma (cid:3)(cid:2)(cid:5) Scenario D of (cid:4)F(cid:0)(cid:22)(cid:6) is an incontestable scenario i(cid:4) for all
d (cid:1) D(cid:1) (cid:4)d is not explainable from (cid:4)F(cid:0)(cid:22)(cid:6)(cid:0)
Proof(cid:6) Scenario S explains (cid:4)D i(cid:27) there is someminimalsubset
(cid:0) (cid:0)
D of D such that F (cid:0)S j(cid:17) (cid:4)D (cid:3)
(cid:0)
The lemma follows from noticing that if D (cid:17) fd(cid:0)(cid:0)(cid:4)(cid:4)(cid:4)(cid:0)dng(cid:5)
F (cid:0)S j(cid:17) (cid:4)(cid:4)d(cid:0) (cid:5)(cid:4)(cid:4)(cid:4)(cid:5)dn(cid:6)
i(cid:27)
F (cid:0)S (cid:0)fd(cid:0)(cid:0)(cid:4)(cid:4)(cid:4)(cid:0)dn(cid:1)(cid:0)g j(cid:17) (cid:4)dn
and the left hand side of each formula is consistent (cid:4)by the mini(cid:11)
(cid:0)
(cid:0)
mality of D (cid:6)(cid:3)
Explanation and Prediction (cid:9)
Thus being part of an incontestable scenario is a local property of in(cid:11)
stances of defaults and so there is a unique incontestable extension(cid:5) de(cid:2)ned
as(cid:14)
Corollary (cid:3)(cid:2)(cid:7) g is incontestably explainable from (cid:4)F(cid:0)(cid:22)(cid:6) i(cid:27) g logically fol(cid:11)
lows from F (cid:0)D where
D (cid:17) fd (cid:14) d is a ground instance of an element of (cid:22) and (cid:4)d is not explainable from (cid:4)F(cid:0)(cid:22)(cid:6)g
For the ground case(cid:5) if we can explain the negation of a default(cid:5) it can be
removed at compile time(cid:3) A new knowledge base can be built by removing
any default from (cid:22) for which we can explain its negation (cid:4)from F and the
initial (cid:22)(cid:6)(cid:5) and then computing logical consequences of the facts and the
remaining defaults (cid:4)i(cid:3)e(cid:3)(cid:5) those for which we cannot explain their negations(cid:6)(cid:3)
Forthenon(cid:11)ground case(cid:5)however(cid:5)thisdoesnotworkaswecannot remove
a default just because the negation of some instance of it is explainable(cid:3) In
this case the set D may be in(cid:2)nite(cid:5) however we can still check explainability
dynamically(cid:3)
(cid:2)(cid:1)(cid:3) Membership in all Extensions
The third response to the question posed in section (cid:1) was to predict the
disjunction p(cid:6)q(cid:3)
We do not predict something if we can just explain it(cid:5) as we may be
able to explain it and its negation(cid:3) It seems wrong to both predict some
proposition and also predict its negation(cid:3) It is also not adequate to predict
some proposition because we can explain it and cannot explain its negation(cid:3)
Consider an example where we can explain a and can also explain (cid:4)a(cid:3) We
do not want to predict a is true(cid:3) Suppose the only rule about g is a (cid:2) g(cid:21) if
we can(cid:19)t predict a(cid:5) we do not want to predict g(cid:5) even though there is no way
to explain (cid:4)g(cid:3) Such considerations lead to the idea of predicting what is in
every extension (cid:4)or(cid:5) equivalently what logically follows from the disjunction
of the maximal scenarios(cid:6)(cid:3)
Inthissectionwediscussdi(cid:27)erentproperties ofsuchprediction(cid:21)insection
(cid:12)(cid:3)(cid:1) we show how it can be implemented(cid:3)
In example (cid:1)(cid:3)(cid:0)(cid:5) we predict
religious(cid:4)dick(cid:6)(cid:5)politically(cid:5)motivated(cid:4)dick(cid:6)(cid:5)
(cid:4)(cid:4)hawk(cid:4)dick(cid:6)(cid:5)supports(cid:5)star(cid:5)wars(cid:4)dick(cid:6)(cid:6)(cid:6)dove(cid:4)dick(cid:6)(cid:6)
Explanation and Prediction (cid:28)
This is the formula which is in all extensions (cid:4)together with the facts(cid:5) it is
equivalent to the disjunction of the extensions(cid:6)(cid:3) Whichever extension is true
in a world(cid:5) this formula will be true in that world(cid:3)
The following theorem gives a characterisation of membership in all ex(cid:11)
tensions(cid:14)
Theorem (cid:3)(cid:2)(cid:8) The following are equivalent(cid:6)
(cid:7)(cid:0) g is in every extension of (cid:4)A(cid:0)H(cid:6)(cid:0)
(cid:8)(cid:0) for all scenarios S of (cid:4)A(cid:0)H(cid:6)(cid:1) there is an explanation of g from (cid:4)A (cid:0)
S(cid:0)H(cid:6)(cid:0)
(cid:9)(cid:0) there does not exist a scenario S of (cid:4)A(cid:0)H(cid:6) such that there is no expla(cid:5)
nation of g from (cid:4)A(cid:0)S(cid:0)H(cid:6)(cid:0)
(cid:10)(cid:0) there is a set E of (cid:2)(cid:11)nite(cid:3) explanations of g from (cid:4)A(cid:0)H(cid:6) such that there
is no scenario S of (cid:4)A(cid:0)H(cid:6) inconsistent with every element of E(cid:0)
(cid:12)(cid:0) there is an explanation D of g(cid:1) and if there exists d (cid:1) D such that (cid:4)d
is explainable by E(cid:1) then g is in every extension of (cid:4)A(cid:0)E(cid:0)H(cid:6)(cid:0)
Proof(cid:6) (cid:1) (cid:2) (cid:0)(cid:3) If g is explainable from all scenarios(cid:5) it is
explainable from all maximal scenarios(cid:5) that is it is in every ex(cid:11)
tension(cid:3)
(cid:1) (cid:7) (cid:13)(cid:3) These are rewritings of the same statement(cid:3)
(cid:20) (cid:2) (cid:1)(cid:3) Suppose (cid:20) holds(cid:5) and there is a scenario S from which g
is not explainable(cid:3) Each E (cid:1) E is inconsistent with S (cid:4)otherwise
E (cid:0)S is an explanation of g from (cid:4)A(cid:0)S(cid:0)H(cid:6)(cid:6)(cid:3)
(cid:0) (cid:2) (cid:20)(cid:3) Suppose (cid:0) holds(cid:3) The set of all maximal scenarios
has the property given in (cid:20) (cid:4)except the (cid:2)nite membership(cid:6)(cid:3) By
the compactness theorem of the (cid:2)rst order predicate calculus
(cid:7)Enderton(cid:8)(cid:1)(cid:10) there is a set E composed of (cid:2)nite subsets of the
maximal scenarios which imply g(cid:3) If some S were inconsistent
with all elements of E it would be inconsistent with the maximal
scenarios(cid:5) and we know such an S cannot exist(cid:3) So E is a set
which satis(cid:2)es (cid:20)(cid:3)
Explanation and Prediction (cid:0)(cid:18)
(cid:20) (cid:2) (cid:12)(cid:3) Suppose (cid:20) holds(cid:5) the set E iscountable (cid:4)as itis a subset of
the set of (cid:2)nite strings in a language with countable generators(cid:6)(cid:3)
Let D be the minimumelementof E according to some ordering(cid:3)
We know A(cid:5)D j(cid:17) g(cid:3) As g is in every extension of A(cid:0)H it is in
every extension of A(cid:5)E(cid:0)H(cid:3)
(cid:12) (cid:2) (cid:1)(cid:3) Suppose (cid:12) holds and there is some scenario S such that
g is not explainable from S(cid:3) D is inconsistent with S (cid:4)otherwise
S (cid:0)D is an scenario of S(cid:0)H which explains g(cid:6)(cid:5) so there is some
(cid:0) (cid:0) (cid:0)
d (cid:1) D which follows from consistent S (cid:17) S (cid:0)D where D (cid:8) D
(cid:0)
andsoby(cid:12)(cid:5) g isineveryextensionofS (cid:5)andsoisinoneextension
(cid:0)
(cid:0)
of S (cid:5) a contradiction to g not being explainable from S(cid:3)
This theorem shows that membership in all extensions is also a sceptical
theory of prediction(cid:3)
When predicting what is in all extensions(cid:5) we can think of starting with
all explainable propositions(cid:3) We eliminate a proposition if its negation can
be explained(cid:5) or if its derivation rests on removedpropositions(cid:3) Suppose (cid:3) is
explainable(cid:21) if (cid:4)(cid:3) is explainable (cid:4)by scenario S(cid:6)(cid:5) (cid:3) is not in every extension(cid:3)
If (cid:5) was derived from (cid:3)(cid:5) to be in all extensions (cid:5) must be explainable from
S (cid:3)
Theorem (cid:1)(cid:3)(cid:26) tells us that if g is not in every extension of A(cid:0)(cid:22)(cid:5) there is
some scenario S of A(cid:0)(cid:22)(cid:5) such that g is not explainable from S(cid:0)(cid:22)(cid:3) Based on
defaultsbeing normalityconditions(cid:4)i(cid:3)e(cid:3)(cid:5)conditions that weexpectto betrue
given no evidence to the contrary(cid:6) we cannot rule out S(cid:5) and so we should
not predict g(cid:3)
The di(cid:27)erencebetween predictingwhat is in allextensions and predicting
what is incontestably explainable(cid:5) is that the latter requires one explanation
of the goal which is consistent with all scenarios(cid:5) whereas the former allows
a set of explanations of the goal which must be consistent with all scenarios(cid:3)
Example (cid:3)(cid:2)(cid:9) Suppose we are using the default reasoning system for recog(cid:11)
nition(cid:3) Suppose also that wecan explainPollybeing an emuand also explain
Polly being an ostrich(cid:3) It cannot be both an emu and an ostrich(cid:3)
H (cid:17) f feathered(cid:4)X(cid:6)(cid:5)big(cid:4)X(cid:6)(cid:5)runs(cid:4)X(cid:6) (cid:2) emu(cid:4)X(cid:6)(cid:0)
feathered(cid:4)X(cid:6)(cid:5)big(cid:4)X(cid:6)(cid:5)runs(cid:4)X(cid:6) (cid:2) ostrich(cid:4)X(cid:6)g
F (cid:17) f (cid:3)X (cid:4)(cid:4)emu(cid:4)X(cid:6)(cid:5)ostrich(cid:4)X(cid:6)(cid:6)